The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

Finite element methods for Maxwell's equations.

During the last decade, a big progress has been achieved in the analysis and numerical treatment of Initial Value Problems (IVPs) in Differential Algebraic Equations (DAEs) and Ordinary Differential Equations (ODEs). In spite of the rich variety of results available in the literature, there are still many specific problems that require special attention. Two of such, which are considered in this work, are the optimization of order of accuracy and reduction of cost of functional evaluations of Explicit Runge - Kutta (ERK) methods.

Traditionally, the maximum attainable order p of an s-stage ERK method for advancing the solution of an IVP is such that
p(s)
 4
 
In 1999, Goeken presented an s-stage ERK Method of order p(s)=s +1,s>2.
However, this work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher 
U n and Jonhson [94]; and the classical ERK methods. The order p of 
the new scheme called Multiderivative Explicit Runge-Kutta (MERK) Methods is such that p(s) 2. The stability, convergence and implementation for the optimization of IVPs in DAEs and ODEs systems are also considered.

Numerical solution of initial value problems in differential - algebraic equations

Preparing the books to read every day is enjoyable for many people. However, there are still many people who also don't like reading. This is a problem. But, when you can support others to start reading, it will be better. One of the books that can be recommended for new readers is computer aided analysis of electronic circuits algorithms and computational techniques. This book is not kind of difficult book to read. It can be read and understand by the new readers.

Computer-aided analysis of electronic circuits: Algorithms and computational techniques

Differential algebraic equations (DAEs), including so-called descriptor systems, began to attract significant research interest in applied and numerical mathematics in the early 1980s, no more than about three decades ago. In this relatively short time, DAEs have become a widely acknowledged tool to model processes subjected to constraints, in order to simulate and to control processes in various application fields such as network simulation, chemical kinematics, mechanical engineering, system biology. DAEs and their more abstract versions in infinite-dimensional spaces comprise a great potential for future mathematical modeling of complex coupled processes. The purpose of the book is to expose the impressive complexity of general DAEs from an analytical point of view, to describe the state of the art as well as open problems and so to motivate further research to this versatile, extra-ordinary topic from a broader mathematical perspective. The book elaborates a new general structural analysis capturing linear and nonlinear DAEs in a hierarchical way. The DAE structure is exposed by means of special projector functions. Numerical integration issues and computational aspects are treated also in this context.

Differential-Algebraic Equations: A Projector Based Analysis

The development of integrated circuits requires powerful numerical simulation programs. Naturally, there is no method that treats all the different kinds of circuits successfully. The numerical simulation tools provide reliable results only if the circuit model meets the assumptions that guarantee a successful application of the integration software. Owing to the large dimension of many circuits (about 107 circuit elements) it is often difficult to find the circuit configurations that lead to numerical difficulties. In this paper, we analyse electric circuits with respect to their structural properties in order to give circuit designers some help for fixing modelling problems if the numerical simulation fails. We consider one of the most frequently used modelling techniques, the modified nodal analysis (MNA), and discuss the index of the differential algebraic equations (DAEs) obtained by this kind of modelling. Copyright © 2000 John Wiley & Sons, Ltd.

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Structural analysis of electric circuits and consequences for MNA

Differential Algebraic Systems In order to treat coupled systems of partial differential equations (PDEs) and differential-algebraic equations (DAEs) in a systematic way we shall study abstract differential algebraic systems (ADASs) of the following form A d dt D(u(t), t) + B(u(t), t) = 0 for t ∈ [t0, T ]. (4.1) This equation is to be understood as an operator equation with operators A, D(·, t) and B(·, t) acting in real Hilbert spaces. More precisely, let X, Y , Z ⊆ Z̃ be Hilbert spaces and A : Z̃ → Y, D(·, t) : X → Z, B(·, t) : X → Y. (4.2) In [Mär02a, LMT01], the case Z = Z̃ has been considered. This is enough for the classical formulation of the coupled system (see Section 3.5). However, for the generalized formulation, we need the more general case Z ⊆ Z̃ (see Section 3.6). Such systems with A and D being invertible have already been studied in [GGZ74]. However, the classical formulation as well as the generalized formulation of the coupled system lead to abstract differential algebraic systems with operators A or D(·, t) that are not invertible on the whole time interval. Such systems are also called singular (see e.g. [Cam80, Cam82] for the finite dimensional case) or degenerate differential equations (see e.g. [Kur93, FY99]). 66 Abstract Differential Algebraic Systems In [FY99], systems of the form (4.1) with linear operators have been studied. More precisely, initial value systems of the form A d dt Du + Bu = Ag(t) for t ∈ (0, T ], (4.3) Du(0) = v0 ∈ imD, (4.4) with X = Y = Z, D = M and A = I are treated by a semigroup approach. I denotes the identity operator. The system (4.3)-(4.4) is reduced to multivalued differential equations of the form dv dt ∈ Av + f(t) for t ∈ (0, T ], (4.5) v(0) = v0, (4.6) where A := −BD. Here, the operator D is defined as (multi-valued) function satisfying Dv = {u ∈ DD : Du = v} for all v ∈ imD. with DD being the definition domain of D. The existence and uniqueness of classical solutions of (4.3)-(4.4) satisfying Du ∈ C([0, T ]; X) and Bu ∈ C([0, T ]; X) is shown provided that g ∈ C([0, T ]; X), Bu(0) ∈ imA, Re(−Bu|Du)X ≤ β‖Du‖ 2 X , for all u ∈ DB ⊆ DD (4.7) as well as the operator λ0AD + B : DB → X is bijective for some λ0 > β. (4.8) A similar result is obtained for systems of the form (4.3)-(4.4) with A = D or D = I. This approach via multi-valued differential equations concentrates on the dynamic part of the system. It is limited to systems with special constraints. So, for instance, condition (4.7) implies Re(Bw,Dv) = 0 for all w ∈ kerD ∩ DB, v ∈ DD. Together with condition (4.8), we obtain, for the finite dimensional case, a DAE of index 1 with the constraint (BQ)Bu = (BQ)g(t)

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Coupled Systems of Differential Algebraic and Partial Differential Equations in Circuit and Device Simulation

In electric circuit simulation we are confronted with highly nonlinear DAEs with low smoothness properties. They may have index 2 but they do not belong to the class of Hessenberg form systems that are well understood. The classic and the charge-oriented modified analysis are shown to lead to the same DAE-index if the circuit models satisfy some natural assumptions.



We present a topological criteria for calculating the index. This makes it possible to determine the index also for high-dimensional circuit equation systems.

Topological index‐calculation of DAEs in circuit simulation

We explore the Tractability Index of Differential Algebraic Equations (DAEs) that emerge in the simulation of gas transport networks. Depending on the complexity of the network, systems of index 1 or index 2 can arise. It is then shown that these systems can be rewritten as Ordinary Differential Equations (ODEs). We furthermore apply Model Order Reduction (MOR) techniques such as Proper Orthogonal Decomposition (POD) to a network of moderate size and complexity and show that one can reduce the system size significantly.

Caren Tischendorf

Papers

Differential-Algebraic Equations: A Projector Based Analysis

Structural analysis of electric circuits and consequences for MNA

Coupled Systems of Differential Algebraic and Partial Differential Equations in Circuit and Device Simulation

Topological index‐calculation of DAEs in circuit simulation

Model Order Reduction of Differential Algebraic Equations Arising from the Simulation of Gas Transport Networks